# The Maths Behind The Hundred-Year Flood

Some unfortunate news reporter, wrapped in an inadequate waterproof and a flimsy hat, is standing perilously close to a sea-wall as the waves crash in. Shouting over the raging storm, he states that the extreme weather affecting the town is forecast to cause a "hundred-year flood". Perhaps distracted by emergency services suggesting he move to a safer location, he fails to explain what he means by this.

### What is a hundred-year flood?

The exceedance probability of a flood is how likely you are to see flooding "this bad or worse" in any given year. Where I live in Weymouth, you can almost guarantee that the Swannery car park will be flooded at least once a year, so the exceedance probability of a Swannery car Park Flood is close to 100%. I live near the top of a hill, which - touch wood - never ever floods, so the exceedance probability of a Colin's House Flood is effectively 0%. Most places, though, have an exceedance probability somewhere in between.

A hundred-year flood is one with an exceedance probability of less than 1% - on average, you'd expect to see flooding this bad about once a century.

There's nothing particularly special about the figure of 100 years; you can think just as easily about 20-year floods (an exceedance probability of less than 5%), 1000-year floods (less than 0.1%) or any-number-you-like-year floods - an x-year flood requires an exceedance probability below 100/x%.

But while 100 years is an arbitrary cut-off, it's a significant one: insurance companies often use maps showing areas at risk of 100-year floods in order to determine risk premiums.

### So hundred-year floods happen every hundred years?

Not so fast. A hundred-year flood (or worse) occurs once every hundred years on average. But that doesn't mean you can set your calendar by it.

Let's assume, for the moment, than the climate is fixed and unchanging and effectively rolls a 100-sided die every year, and throws up a 100-year flood if it rolls a 1. If it does this 100 times, what's the probability of getting at least one 100-year flood?

For this, we need the binomial distribution. The probability of getting no 100-year floods is (0.99)100, which works out to be about 36.8% - so any given century has a roughly 63% chance of a hundred-year flood.

On the flip side, the probability of having exactly one 100-year flood is 100 (0.99)99 x 0.01, only a tiny bit bigger (37.0%) - which means that in the remaining 26.2% of centuries, more than a quarter, you expect to get at least two hundred-year floods.

### We seem to have had a lot of hundred-year floods here recently.

Surely you're not suggesting that... the climate might be changing?! I thought that was a Chinese conspiracy. Let me put this tin-foil hat down and do some statistics instead.

One way to test the evidence on climate change is to do a hypothesis test on the data, which has a sense of a criminal trial about it. You only want to conclude that something has changed if the data would be extremely unlikely had it not changed - you show, beyond reasonable doubt, that your assumptions must be wrong.

For example, suppose I tossed a coin I thought was fair 25 times and it came up heads every time. The odds against that run of results, given that it's a fair coin, are enormous - about one in 33.5 million. Under the circumstances, I'd be justified in saying "I no longer believe this coin is fair."

There is a process for conducting a hypothesis test: * You make a null hypothesis - generally, this assumption is that nothing has changed. For the experiment we're about to do, this will be "The probability of a flood this bad in a given year is 1%". * Our alternate hypothesis is that things have changed somehow. Here, we want to know whether the probability has increased, so it will be "the probability of a flood this bad in a given year is greater than 1%". * You then find the probability of the data (or a more extreme result), assuming the null hypothesis is true. * If this probability is less than a given threshold - often 0.05 - you reject the null hypothesis in favour of the alternate hypothesis. Otherwise you accept the null hypothesis.

(Like a not guilty verdict in a trial, accepting the null hypothesis doesn't necessarily mean the alternate hypothesis is false - just that it hasn't been adequately proved.)

So, looking at the data for a fictional riverside town, we find there have been three hundred-year floods in the last 20 years. What's the probability of having three or more such floods, if the probability is really 0.01 in each year?

Again, the binomial expansion comes to the rescue (working below): the probability of 0, 1 or 2 100-year floods in 20 years is 99.9%, so the probability of three or more is 0.1%. This is smaller than our 5% threshold, so - in this made-up experiment - we would reject the null hypothesis that floods this bad occur on average every hundred years in favour of the alternate hypothesis, that they occur more frequently.

This - an experiment with made-up data, I should emphasise - would suggest that severe flooding was becoming more frequent. There could be all sorts of reasons for that, not necessarily related to climate change - from changes in the local flood plain to problems with the drainage to a collapse river wall that hasn't been repaired.

So this experiment wouldn't, on its own, allow you to conclude that climate change was real. However, repeating the experiment with other locations and other weather patterns would allow us to build up a convincing statistical argument for it (which is, of course, exactly what the Met Office, the IPCC and countless others have been doing over the last few decades.)

Now, over to Gail with the weather.

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Working for the hypothesis test.

To find P(X≥3), we can work out 1 - P(X≤2). To do that, we can find the probability that X is 0, 1 or 2 directly.

P(X=0) = 1 * 0.99^20 * 0.01^0 ~ 0.8179 P(X=1) = 20 * 0.99^19 * 0.01^1 ~ 0.1652 P(X=2) = 190 * 0.99^18 * 0.01^2 ~ 0.0159

Therefore, P(X≤2) = 0.9990 and P(X≥3) = 0.0010.

By Colin Beveridge